C₂×#32983 #6629131

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6342 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16,17,19)(18,20,22,21);(8,11)(12,14)(16,19)(20,21);(7,9,10,13)(8,12,11,14)(16,20)(18,22)(19,21);(4,6)

maximal subgroup of	C₂×#6577292, C₂×#6629278, C₂×#6575026, …
maximal normal subgroup of	C₂×#6577292, C₂×#6629278, C₂×#6575026, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536327);